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Alright, I think I should write my 2 cents here:

Obviously $Spec(\mathbb{Q})$ and $\mathbb{A}_K$ are not directly analogous, but they do appear to be in relation to this problem. It seems that they are related through the intermediary $Spec(F)$ where $F$ is a function field over an algebraically closed field. Shafarevich's conjecture holds for $Spec(F)$ (this is an earlier result).

If we take any open smooth affine curve, $C$, over an algebraically closed field of positive characteristic; and use the result that $\pi_1^c(C)$ is pro-finite free. Every abelian covering of $C$ will give an abelian extension of $\kappa(C)$ ($C$'s function field). However, there's no reason to think we get all abelian extensions of $\kappa(C)$ that way. However $\kappa(C)$ is also $\kappa(D)$ for different smooth affine curves, so may use their abelian unramified covers.

To make some order of this, start with an abelian extension of $\kappa(C)$, $L$. We may take $C$'s normalization in $L$. This may be branched at some points in $C$, but we may discard those. So it seems that any abelian extension of $\kappa(C)$ comes from an abelian unramified cover of some possibly different smooth affine curve whose function field is $\kappa(C)$.

It seems, however, extraordinary to expect that since $\pi_1^c(Spec(\kappa(C)))$ is pro-finite free, $\pi_1^c(C)$ should be; for any affine curve $C$. Is there some secret motivation for thinking this that I'm missing?

1

Alright, I think I should write my 2 cents here:

Obviously $Spec(\mathbb{Q})$ and $\mathbb{A}_K$ are not directly analogous, but they do appear to be in relation to this problem. It seems that they are related through the intermediary $Spec(F)$ where $F$ is a function field over an algebraically closed field. Shafarevich's conjecture holds for $Spec(F)$ (this is an earlier result).

If we take any open affine curve, $C$, over an algebraically closed field of positive characteristic; and use the result that $\pi_1^c(C)$ is pro-finite free. Every abelian covering of $C$ will give an abelian extension of $\kappa(C)$ ($C$'s function field). However, there's no reason to think we get all abelian extensions of $\kappa(C)$ that way. However $\kappa(C)$ is also $\kappa(D)$ for different affine curves, so may use their abelian unramified covers.

To make some order of this, start with an abelian extension of $\kappa(C)$, $L$. We may take $C$'s normalization in $L$. This may be branched at some points in $C$, but we may discard those. So it seems that any abelian extension of $\kappa(C)$ comes from an abelian unramified cover of some possibly different affine curve whose function field is $\kappa(C)$.

It seems, however, extraordinary to expect that since $\pi_1^c(Spec(\kappa(C)))$ is pro-finite free, $\pi_1^c(C)$ should be; for any affine curve $C$. Is there some secret motivation for thinking this that I'm missing?